Compound channel single friction in hydraulic flume

Experimental set up

A compound channel setup was used for this case study. The paper including all information about the experience was found in Proust and Nikora (2020), data are available in PROUST and NIKORA (2025).

• Length: 18 m
• Width: 3 m
• Slope: 0.0011
• Main channel: 1 m wide rectangular glass bed
• Floodplain: two 1 m wide flat rough-surface covered with dense artificial grass (consisting of 1 mm wide and 5 mm high thin rigid blades, with a density of 256 blades per square centimeter).
• Vertical distance from the main channel to the floodplain: 0.117 m
• Flow conditions: uniform flow with a total discharge of 114 L/s, of which 8 L/s passed through the floodplain
• Water level measurements: taken at spatial intervals of 0.3 to 1 m along the streamwise direction, at transverse positions at y=0.3 and 0.7 m on the right-hand floodplain and at y=1.2, 1.5 and 1.8 m in the main channel.

Main features for the estimation

• A friction coefficient is estimated in the main channel and other one in the floodplain.
• A constant longitudinal friction is considered.
• No more polynomial degrees have been performed.
• A single event is used for the calibration. The observed WSE is taken from the average of the measurements in the main channel and the floodplain.
• Experiments performed have just changed the observational data to answer some questions about the identifiability of the friction coefficients in the main channel and the floodplain.
• In the real case, upstream discharge was divided between the main channel and the floodplain. In the model, a total discharge is considered without any distinction.
• Measurement of WSE were taken in the main channel and the floodplain. For this case, an average value was calculated for each cross-section to compare with the simulations. All averaged depth values were converted to elevation using the downstream as the reference datum.
• Uncertainty of the observations was calculated using a regression function passing through the water depth and then a standard deviation was calculated for all observations.
• No measurements of the downstream threshold were reported. Therefore, the last available measurements were taken to define the downstream boundary condition in the model.

Laboratory measurements

Take a look about all the measurements:

Experiment: `1_WSE_floodplain_realistic_uncertainty

Here the experiment with the observational data obtained in the laboratory,averaging the values of the WSE.

Calibration data

A sample was taken of all measurements shown previously:

Check MCMC

All MCMC samples:

MCMC cooked:

Corelation plot of MCMC cooked:

Check summary

Zoom into the MAP and standard deviation of the error model: WSE in mm, discharge in m3/s, velocity in m/s kmin and kmoy in m1/3/s.

	a0_min	a0_flood	Y1_intercept	Y2_intercept	Y3_intercept	Y4_intercept	Y5_intercept
N	2001.0000000	2001.000000	2.00100e+03	2001.000000	2001.000000	2001.000000	2001.000000
Minimum	107.4760000	23.356700	1.50000e-06	5.350170	5.473510	7.743410	6.253130
Maximum	130.0730000	46.343400	1.75630e-03	17.402300	22.259300	30.763900	27.444300
Range	22.5970000	22.986700	1.75480e-03	12.052100	16.785800	23.020500	21.191200
Mean	120.2120000	32.501500	2.06800e-04	10.045000	10.231900	15.379100	15.357700
Median	119.6200000	32.874100	1.71700e-04	9.793080	10.025000	15.268400	15.098500
Q10%	114.0310000	26.350600	3.29000e-05	7.671800	7.533980	11.570400	11.315500
Q25%	116.5180000	28.121200	6.60000e-05	8.672890	8.635640	13.172900	12.937500
Q75%	124.5030000	36.109800	3.06300e-04	11.206600	11.669000	17.080600	17.506600
Q90%	126.5360000	38.714200	4.31000e-04	12.853100	13.112000	19.466900	19.593900
St.Dev.	4.8327100	4.779110	1.78100e-04	1.971020	2.200820	3.108970	3.275740
Variance	23.3551000	22.839900	0.00000e+00	3.884940	4.843600	9.665710	10.730500
CV	0.0402015	0.147042	8.61514e-01	0.196220	0.215094	0.202156	0.213297
Skewness	0.0730142	0.104323	2.22869e+00	0.635261	0.571186	0.611240	0.543355
Kurtosis	-0.9667830	-0.821106	1.15867e+01	0.351366	0.620198	1.211530	0.343737
MaxPost	119.8560000	32.305200	6.37000e-05	10.483900	9.980640	13.988500	15.570500

	a0_min	a0_flood	Y1_intercept	Y2_intercept	Y3_intercept	Y4_intercept	Y5_intercept
St.Dev.	4.83271	4.77911	0.1781380	1.97102	2.20082	3.10897	3.27574
MaxPost	119.85600	32.30520	0.0636728	10.48390	9.98064	13.98850	15.57050

Estimation of the friction coefficients

In the main channel:

Estimation of the friction coefficients

In the floodplain:

Residuals

in terms of WSE

In terms of discharge

Notes:

• High negative corelation between a0_kmin and a0_kmoy. They can be interchangeable and the result will be the same with a weighting or factor to compensate.

Question 1:

How to reduce the corelation between a0_kmin and a0_kmoy using only a single event?

• First proposal: reduce the uncertainty in the observational data (case 1_WSE_floodplain_null_uncertainty)
• Second proposal: add a single pseudo-observation of the friction in the main channel with very low uncertainty (case 1_WSE_floodplain_1_Kmin_null_uncertainty)
• Third proposal: add several pseudo-observations of the friction in the main channel with a realistic uncertainty (case 1_WSE_floodplain_several_Kmin)
• Fourth proposal: similar as the second proposal but now the pseudo-observation is taken in the floodplain (case 1_WSE_floodplain_1_Kmoy_null_uncertainty)

Main conclusions:

The conclusions are presented before describing the experiments, in order to highlight the key findings of the study.

In laboratory measurements:

• Water depth measurements presented a discrepancy between measurements in the main channel and the floodplain. Averaging them could already introduce some errors during modelling.
• A light backwater wave was observed from downstream to upstream. Indeed, the hydraulic control is the weir located downstream which is controlling the flow dynamics.
• The height of the weir is not reported. Therefore, the closest observed water level was chosen as the downstream boundary condition, introducing non-negligible errors in the modeling.

Results of experiments:

• High corelations between the friction in the main channel and the floodplain were detected.
• To reduce this corelation, it is necessary to get more information about the friction at least from one of them, WSE observation with low uncertainty to restreint the possible combinations of parameters.
• The method is able to estimate the parameters, but special attention should be paid to the selection of observational data in order to avoid very high parametric uncertainties.
• To reproduce the observed WSE, the friction values in either the main channel or the floodplain must be set to unrealistic levels according to the literature.

Underlying questions:

1. The method could identify the friction in the main channel and the floodplain using a single event in floodplain?

   Answer: 1_WSE_floodplain_realistic_uncertainty experiment, presented previously, shows calibration using a single event in floodplain lead to estimate a high number of good combinations for reproducing the observed WSE. So, it is possible but parametric uncertainty is huge due to the strong negative corelation

2. How to reduce the corelation between kmin/kmoy ?

   Answer: Taking a pseudo-observation of the friction either in main channel or floodplain with low uncertainty to reduce the number of combinations of the parameters.

3. A minimal number of events (2) to identify the friction in the main channel and floodplain? Should one of the events flow only in the main channel?

   Answer: This experiment must be performed in a synthetic experiment to conclude about it.

Experiment: 1_WSE_floodplain_null_uncertainty

Reduce the uncertainty in the observational data

Calibration data

A sample was taken of all measurements shown previously:

Check MCMC

All MCMC samples:

MCMC cooked:

Corelation plot of MCMC cooked:

Check summary

Zoom into the MAP and standard deviation of the error model: WSE in mm, discharge in m3/s, velocity in m/s kmin and kmoy in m1/3/s.

	a0_min	a0_flood	Y1_intercept	Y2_intercept	Y3_intercept	Y4_intercept	Y5_intercept
N	2001.0000000	2001.000000	2.00100e+03	2001.000000	2001.000000	2001.000000	2001.000000
Minimum	107.4770000	24.728800	2.28700e-04	5.858510	5.162780	7.256530	8.535670
Maximum	128.5200000	47.267600	1.02860e-03	19.895200	19.534400	26.391600	25.783100
Range	21.0430000	22.538800	7.99900e-04	14.036700	14.371600	19.135100	17.247400
Mean	117.0700000	35.709300	4.68000e-04	10.422800	10.340100	15.183500	15.256800
Median	116.6180000	35.924800	4.40500e-04	10.168100	10.021500	15.076100	14.853100
Q10%	111.3130000	29.928700	3.08800e-04	7.961250	7.902230	11.555500	11.659700
Q25%	114.0760000	31.904100	3.64800e-04	9.012630	8.877260	13.179400	13.159800
Q75%	120.6260000	38.717500	5.48300e-04	11.564200	11.582200	17.017600	17.146600
Q90%	122.3280000	42.365800	6.64100e-04	13.277000	13.104500	18.888700	19.327900
St.Dev.	4.3978600	4.729290	1.39500e-04	2.099750	2.128930	2.922280	3.026050
Variance	19.3412000	22.366200	0.00000e+00	4.408930	4.532330	8.539750	9.156990
CV	0.0375661	0.132438	2.98147e-01	0.201457	0.205891	0.192464	0.198341
Skewness	0.1696110	0.105879	8.07301e-01	0.714238	0.737369	0.450935	0.561829
Kurtosis	-0.6100620	-0.633168	4.67690e-01	0.821512	1.008740	0.474089	0.139611
MaxPost	118.9850000	33.238100	3.34500e-04	9.791470	9.998730	14.983100	14.560200

	a0_min	a0_flood	Y1_intercept	Y2_intercept	Y3_intercept	Y4_intercept	Y5_intercept
St.Dev.	4.39786	4.72929	0.139538	2.09975	2.12893	2.92228	3.02605
MaxPost	118.98500	33.23810	0.334454	9.79147	9.99873	14.98310	14.56020

Estimation of the friction coefficients

In the main channel:

Estimation of the friction coefficients

In the floodplain:

Residuals

in terms of WSE

In terms of discharge

Notes:

• From the first case, the uncertainty is already low, thus reducing the uncertainty in the observational data does not help to break the corelation between the parameters.
• To assess the identifiability of the parameters, non uncertainty in observation is considered for the rest of the experiments.

Experiment: 1_WSE_floodplain_1_Kmin_null_uncertainty

Add a single pseudo-observation of the friction in the main channel with very low uncertainty

Calibration data

The WSE observation is considered without uncertainty as the previous example. Below the calibration data is shown:

event	reach	x	t	WSE	Discharge	Velocity	Y_Kmin	Y_Kmoy	Yu_z	Yu_Q	Yu_V	Yu_Kmin	Yu_Kmoy
1	1	3.50	3420	0.1643105	-9999	-9999	-9999	-9999	0	-9999	-9999	-9999	-9999
1	1	5.00	3420	0.1616392	-9999	-9999	-9999	-9999	0	-9999	-9999	-9999	-9999
1	1	6.50	3420	0.1602367	-9999	-9999	100	-9999	0	-9999	-9999	0	-9999
1	1	7.00	3420	0.1603004	-9999	-9999	-9999	-9999	0	-9999	-9999	-9999	-9999
1	1	7.50	3420	0.1590004	-9999	-9999	-9999	-9999	0	-9999	-9999	-9999	-9999
1	1	9.25	3420	0.1575192	-9999	-9999	-9999	-9999	0	-9999	-9999	-9999	-9999
1	1	11.75	3420	0.1552395	-9999	-9999	-9999	-9999	0	-9999	-9999	-9999	-9999
1	1	12.50	3420	0.1543078	-9999	-9999	-9999	-9999	0	-9999	-9999	-9999	-9999
1	1	13.00	3420	0.1539768	-9999	-9999	-9999	-9999	0	-9999	-9999	-9999	-9999

Check MCMC

All MCMC samples:

MCMC cooked:

Corelation plot of MCMC cooked:

Check summary

Zoom into the MAP and standard deviation of the error model: WSE in mm, discharge in m3/s, velocity in m/s kmin and kmoy in m1/3/s.

	a0_min	a0_flood	Y1_intercept	Y2_intercept	Y3_intercept	Y4_intercept	Y5_intercept
N	2.00100e+03	2001.0000000	2.00100e+03	2001.000000	2001.000000	2.00100e+03	2001.000000
Minimum	9.99974e+01	52.1071000	2.09000e-04	5.782940	5.750480	7.60000e-06	7.959880
Maximum	1.00002e+02	63.8039000	1.97570e-03	18.754200	19.024200	6.41560e-03	25.378100
Range	4.60000e-03	11.6968000	1.76670e-03	12.971300	13.273700	6.40800e-03	17.418200
Mean	1.00000e+02	59.3397000	4.64100e-04	10.128900	10.344600	7.17200e-04	15.158400
Median	1.00000e+02	59.3257000	4.21200e-04	9.874710	10.137300	4.06800e-04	14.991500
Q10%	1.00000e+02	57.8182000	3.25900e-04	7.646060	7.841480	1.06200e-04	11.398700
Q25%	1.00000e+02	58.6624000	3.70100e-04	8.614600	8.887270	2.06400e-04	13.211300
Q75%	1.00000e+02	60.0413000	5.21700e-04	11.302100	11.589900	8.40100e-04	16.985500
Q90%	1.00000e+02	60.8575000	6.33300e-04	12.911600	12.932400	1.66080e-03	18.978500
St.Dev.	5.14600e-04	1.2504500	1.70500e-04	2.152070	2.137930	8.37900e-04	2.945500
Variance	3.00000e-07	1.5636100	0.00000e+00	4.631390	4.570740	7.00000e-07	8.675970
CV	5.10000e-06	0.0210727	3.67430e-01	0.212468	0.206672	1.16825e+00	0.194314
Skewness	5.13807e-01	-0.6787840	3.88415e+00	0.788049	0.673140	2.61497e+00	0.368663
Kurtosis	1.27389e+01	4.1952700	2.57040e+01	0.852897	0.802783	8.41432e+00	0.186513
MaxPost	1.00000e+02	59.6315000	3.47900e-04	9.850530	10.173900	2.07500e-04	14.566800

	a0_min	a0_flood	Y1_intercept	Y2_intercept	Y3_intercept	Y4_intercept	Y5_intercept
St.Dev.	5.146e-04	1.25045	0.170512	2.15207	2.13793	0.0008379	2.9455
MaxPost	1.000e+02	59.63150	0.347862	9.85053	10.17390	0.0002075	14.5668

Estimation of the friction coefficients

In the main channel:

Estimation of the friction coefficients

In the floodplain:

Residuals

in terms of WSE

In terms of discharge

Notes:

• By forcing a observation very tight in the main channel and keeping a polynomial degree of 0, indicates that the estimation of the friction in the main channel is already known before the calibration. In other words, in a constant longitudinal friction, with a very tight observation of the fricion coefficient in the main channel or floodplain reduce the number of parameters to estimate of 1.
• No more corelation is observed but this case is not realistic, as the friction coefficient in the main channel is not known accurately.
• By fixing the friction coefficient in the main channel, the estimation of the friction coefficient in the floodplain can be better identified

Experiment: 1_WSE_floodplain_several_Kmin

Add several pseudo-observations of the friction in the main channel with a realistic uncertainty

Calibration data

event	reach	x	t	WSE	Discharge	Velocity	Y_Kmin	Y_Kmoy	Yu_z	Yu_Q	Yu_V	Yu_Kmin	Yu_Kmoy
1	1	3.50	3420	0.1643105	-9999	-9999	100	-9999	0	-9999	-9999	3	-9999
1	1	5.00	3420	0.1616392	-9999	-9999	100	-9999	0	-9999	-9999	3	-9999
1	1	6.50	3420	0.1602367	-9999	-9999	100	-9999	0	-9999	-9999	3	-9999
1	1	7.00	3420	0.1603004	-9999	-9999	100	-9999	0	-9999	-9999	3	-9999
1	1	7.50	3420	0.1590004	-9999	-9999	100	-9999	0	-9999	-9999	3	-9999
1	1	9.25	3420	0.1575192	-9999	-9999	100	-9999	0	-9999	-9999	3	-9999
1	1	11.75	3420	0.1552395	-9999	-9999	100	-9999	0	-9999	-9999	3	-9999
1	1	12.50	3420	0.1543078	-9999	-9999	100	-9999	0	-9999	-9999	3	-9999
1	1	13.00	3420	0.1539768	-9999	-9999	100	-9999	0	-9999	-9999	3	-9999

Check MCMC

All MCMC samples:

MCMC cooked:

Corelation plot of MCMC cooked:

Check summary

Zoom into the MAP and standard deviation of the error model: WSE in mm, discharge in m3/s, velocity in m/s kmin and kmoy in m1/3/s.

	a0_min	a0_flood	Y1_intercept	Y2_intercept	Y3_intercept	Y4_intercept	Y5_intercept
N	2.00100e+03	2001.0000000	2.00100e+03	2001.000000	2001.000000	2.00100e+03	2001.000000
Minimum	9.69799e+01	51.4427000	2.01100e-04	4.723730	5.544260	2.64000e-05	7.488510
Maximum	1.02902e+02	67.3106000	1.48350e-03	21.905700	18.814600	1.18611e-02	28.047800
Range	5.92210e+00	15.8679000	1.28240e-03	17.182000	13.270300	1.18347e-02	20.559300
Mean	1.00071e+02	59.1494000	4.70400e-04	10.191200	10.183600	1.57470e-03	15.265000
Median	1.00066e+02	59.1092000	4.33100e-04	9.885030	9.977880	9.85900e-04	15.013200
Q10%	9.88757e+01	56.4398000	3.04300e-04	7.597470	7.684120	2.37900e-04	11.680100
Q25%	9.94580e+01	57.9080000	3.58700e-04	8.524780	8.800370	5.04400e-04	13.165500
Q75%	1.00756e+02	60.5620000	5.35600e-04	11.540700	11.523600	1.82330e-03	17.117400
Q90%	1.01242e+02	61.8728000	6.65200e-04	13.106700	12.806200	3.70820e-03	19.204900
St.Dev.	9.74019e-01	2.2025400	1.62900e-04	2.254560	2.015270	1.70100e-03	3.113380
Variance	9.48712e-01	4.8511700	0.00000e+00	5.083040	4.061320	2.90000e-06	9.693120
CV	9.73330e-03	0.0372369	3.46397e-01	0.221227	0.197894	1.08017e+00	0.203956
Skewness	2.04720e-01	-0.1253350	1.76786e+00	0.838544	0.512311	2.23643e+00	0.556226
Kurtosis	2.70351e-01	0.7538870	5.10156e+00	1.458950	0.551952	5.63228e+00	0.690285
MaxPost	1.00066e+02	59.3612000	3.42000e-04	8.753080	9.791300	5.72200e-04	14.862800

	a0_min	a0_flood	Y1_intercept	Y2_intercept	Y3_intercept	Y4_intercept	Y5_intercept
St.Dev.	0.974019	2.20254	0.162940	2.25456	2.01527	0.0017010	3.11338
MaxPost	100.066000	59.36120	0.341959	8.75308	9.79130	0.0005722	14.86280

Estimation of the friction coefficients

In the main channel:

Estimation of the friction coefficients

In the floodplain:

Residuals

in terms of WSE

In terms of discharge

Notes:

• Several pseudo-observations of the friction in the main channel lead to target a set of combinations of the friction coefficients in the main channel and the floodplain leading to reproduce the observed WSE.
• The corelation between te parameters is reduced but not completely removed.

Experiment: 1_WSE_floodplain_1_Kmoy_null_uncertainty

Similar as the second proposal but now the pseudo-observation is taken in the floodplain

Calibration data

event	reach	x	t	WSE	Discharge	Velocity	Y_Kmin	Y_Kmoy	Yu_z	Yu_Q	Yu_V	Yu_Kmin	Yu_Kmoy
1	1	3.50	3420	0.1643105	-9999	-9999	-9999	-9999	0	-9999	-9999	-9999	-9999
1	1	5.00	3420	0.1616392	-9999	-9999	-9999	-9999	0	-9999	-9999	-9999	-9999
1	1	6.50	3420	0.1602367	-9999	-9999	-9999	33	0	-9999	-9999	-9999	0
1	1	7.00	3420	0.1603004	-9999	-9999	-9999	-9999	0	-9999	-9999	-9999	-9999
1	1	7.50	3420	0.1590004	-9999	-9999	-9999	-9999	0	-9999	-9999	-9999	-9999
1	1	9.25	3420	0.1575192	-9999	-9999	-9999	-9999	0	-9999	-9999	-9999	-9999
1	1	11.75	3420	0.1552395	-9999	-9999	-9999	-9999	0	-9999	-9999	-9999	-9999
1	1	12.50	3420	0.1543078	-9999	-9999	-9999	-9999	0	-9999	-9999	-9999	-9999
1	1	13.00	3420	0.1539768	-9999	-9999	-9999	-9999	0	-9999	-9999	-9999	-9999

Check MCMC

All MCMC samples:

MCMC cooked:

Corelation plot of MCMC cooked:

Check summary

Zoom into the MAP and standard deviation of the error model: WSE in mm, discharge in m3/s, velocity in m/s kmin and kmoy in m1/3/s.

	a0_min	a0_flood	Y1_intercept	Y2_intercept	Y3_intercept	Y4_intercept	Y5_intercept
N	2.00100e+03	2.00100e+03	2.00100e+03	2001.000000	2001.000000	2001.000000	2.00100e+03
Minimum	1.15792e+02	3.29932e+01	2.08100e-04	5.669440	5.187250	7.542560	1.69000e-05
Maximum	1.22672e+02	3.30102e+01	1.33990e-03	17.042600	20.126400	30.747800	8.93260e-03
Range	6.88000e+00	1.70000e-02	1.13170e-03	11.373200	14.939200	23.205200	8.91580e-03
Mean	1.19351e+02	3.30002e+01	4.57400e-04	9.985180	10.344600	15.347600	9.48900e-04
Median	1.19345e+02	3.30000e+01	4.31900e-04	9.845290	10.207600	14.865900	6.33100e-04
Q10%	1.18555e+02	3.29997e+01	3.09100e-04	7.537140	7.828650	11.550500	1.76700e-04
Q25%	1.18880e+02	3.30000e+01	3.57000e-04	8.619260	8.932830	12.960900	3.17700e-04
Q75%	1.19792e+02	3.30004e+01	5.16200e-04	11.098300	11.668900	17.215400	1.21600e-03
Q90%	1.20155e+02	3.30008e+01	6.24800e-04	12.658800	12.998300	20.202400	2.04340e-03
St.Dev.	6.76636e-01	9.15200e-04	1.42900e-04	1.971730	2.031410	3.297080	9.98300e-04
Variance	4.57836e-01	8.00000e-07	0.00000e+00	3.887730	4.126610	10.870700	1.00000e-06
CV	5.66930e-03	2.77000e-05	3.12305e-01	0.197466	0.196373	0.214826	1.05207e+00
Skewness	1.90669e-02	1.87550e-01	1.52533e+00	0.605765	0.473508	0.812506	2.79451e+00
Kurtosis	1.30900e+00	3.11188e+01	3.75694e+00	0.461507	0.563427	1.123140	1.12049e+01
MaxPost	1.19483e+02	3.30000e+01	3.82300e-04	10.050600	8.951850	14.855300	2.04800e-04

	a0_min	a0_flood	Y1_intercept	Y2_intercept	Y3_intercept	Y4_intercept	Y5_intercept
St.Dev.	0.676636	9.152e-04	0.142855	1.97173	2.03141	3.29708	0.0009983
MaxPost	119.483000	3.300e+01	0.382287	10.05060	8.95185	14.85530	0.0002048

Estimation of the friction coefficients

In the main channel:

Estimation of the friction coefficients

In the floodplain:

Residuals

in terms of WSE

In terms of discharge

Notes:

• As in the second proposal, the friction coefficient in the floodplain is already known before the calibration due to the tight uncertainty in the pseudo-observation.
• By fixing this parameters, the estimation shows that the model needs to compensate the friction in the main channel to reproduce the observed WSE.
• The corelation between the parameters is broken but the uncertainty on the pseudo-observation is not realistic.

Bibliography:

PROUST, SEBASTIEN, and VLADIMIR I. NIKORA. 2025. “Dataset of a Laboratory Study on Flows in a Compound Open Channel with Transverse Currents.” Recherche Data Gouv. https://doi.org/10.57745/HJKRYH.

Proust, Sébastien, and Vladimir I. Nikora. 2020. “Compound Open-Channel Flows: Effects of Transverse Currents on the Flow Structure.” Journal of Fluid Mechanics 885 (February): A24. https://doi.org/10.1017/jfm.2019.973.